ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

3.2 Coupled elastic-biphasic model
In order to describe the complete PDL-tooth system, the biphasic model has to be
complemented with elastic model for the tooth. The tooth can be considered a rigid
body with respect to the PDL so that small strain occurs in Ω E , the reference domain of
the tooth. Therefore the momentum equation is solved with a linear elastic law for the
stress tensor σ E . At the interface Γ I between the tooth and the PDL, the following
compatibility conditions should hold:
u S = u E
(σ S +σ F )⋅ν B = −σ E ⋅ν E .
4. TIME DISCRETIZATION AND WEAK FORM
Since we are dealing with a constrained problem, an implicit time discretization is more
suitable since the Lagrange multiplier has no initial condition. Introducing the time step
Δt at the time n the following stationary non-linear problem has to be solved:
−Div(F n S n − J n p n (F −T ) n ) = 0
−J n + ΔtDiv((J n n
k
− nS0 )
γ (C−1 ) n ∇p n ) = −J n−1
Multiplying the first line by a sufficiently smooth vector valued test function v and the
second line by a scalar function q , we obtain the weak formulation after integration by
parts:
(F n S n − J n p n (F −T ) n ∫ ) :V = ∫ t ⋅ v
Ω B
− J n ∫ q − Δt (J n ∫ − nS0) ∫ − ∫ gq
Ω B
Ω B
k n
∂Ω B
γ (C−1 ) n ∇p n ⋅ ∇q = − J n−1 q
ΩB where V = ∇v . The test functions are chosen such that they vanish where a Dirichlet
boundary condition is imposed. On the remaining part of the boundary, t represents the
boundary forces applied on the biphasic body, while g is the net flux of fluid through
the boundary surface. A homogeneous Neumann condition on the boundary indicates
that no liquid is flowing and the total volume is preserved.
5. LINEARISATION AND FINITE ELEMENT DISCRETISATION
The system (13) is a non-linear saddle-point problem and for its solution we use the
Newton method. At every time step, we introduce a subscript j to denote the Newton
n
iterations. As initial guess (uS )0 the displacement at the previous time step is employed.
A similar approach for the pressure led to instabilities and convergence was possible
∂Ω B
(12)
(13)